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Mirrors > Home > ILE Home > Th. List > ecovicom | Unicode version |
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.) |
Ref | Expression |
---|---|
ecovicom.1 | |
ecovicom.2 | |
ecovicom.3 | |
ecovicom.4 | |
ecovicom.5 |
Ref | Expression |
---|---|
ecovicom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovicom.1 | . 2 | |
2 | oveq1 5749 | . . 3 | |
3 | oveq2 5750 | . . 3 | |
4 | 2, 3 | eqeq12d 2132 | . 2 |
5 | oveq2 5750 | . . 3 | |
6 | oveq1 5749 | . . 3 | |
7 | 5, 6 | eqeq12d 2132 | . 2 |
8 | ecovicom.4 | . . . 4 | |
9 | ecovicom.5 | . . . 4 | |
10 | opeq12 3677 | . . . . 5 | |
11 | 10 | eceq1d 6433 | . . . 4 |
12 | 8, 9, 11 | syl2anc 408 | . . 3 |
13 | ecovicom.2 | . . 3 | |
14 | ecovicom.3 | . . . 4 | |
15 | 14 | ancoms 266 | . . 3 |
16 | 12, 13, 15 | 3eqtr4d 2160 | . 2 |
17 | 1, 4, 7, 16 | 2ecoptocl 6485 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cop 3500 cxp 4507 (class class class)co 5742 cec 6395 cqs 6396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fv 5101 df-ov 5745 df-ec 6399 df-qs 6403 |
This theorem is referenced by: addcomnqg 7157 mulcomnqg 7159 addcomsrg 7531 mulcomsrg 7533 axmulcom 7647 |
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